what is the range of cos{√x} where {.} represents fractional part

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Cosine Function Range: The range of the cosine function, cos(θ), for any real number θ is always between -1 and 1, inclusive. This means that for any input value, the output of the cosine function will be within the interval [-1, 1].Fractional Part Function: The fractional part function, {x}, returns the fractional part of x, which is x - ⌊x⌋, where ⌊x⌋ is the greatest integer less than or equal to x. The range of the fractional part function is [0, 1), meaning it includes 0 but does not include 1.Behavior on Interval: Since the fractional part of √x is always between 0 and 1, and cosine is a periodic function with period 2π, we need to consider how the cosine function behaves on the interval [0, 1) to determine the range of cos{√x}.Range Calculation: The cosine function is decreasing on the interval [0, π/2]. Since π/2 is approximately 1.57, which is greater than 1, the interval [0, 1) falls within the decreasing part of the cosine function's first period.Final Range: Because the cosine function is decreasing on [0, π/2] and the fractional part of √x will never reach 1, the range of cos{√x} will be from cos(0) to cos(1), since cos(1) is less than cos(0) due to the decreasing nature of the function in this interval.Calculating the values, we have cos(0) = 1 and cos(1) is approximately 0.5403. Therefore, the range of cos{√x} is from approximately 0.5403 to 1.

what is the range of $\cos{\sqrt{x}}$ where ${.}$ represents fractional part

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what is the range of $\cos{\sqrt{x}}$ where ${.}$ represents fractional part